This invention relates to an apparatus for graphical display of a function of a variable, such apparatus utilizing electronic techniques to provide the display.
A function is a correspondence rule: for any allowable "input" number, the function gives a specific "output" number. The functions most used in applied science are definable by a mathematical expression. In the expression a symbol, such as "x", stands for any input value, and this symbol is therefore referred to as a "variable". Another symbol such as "y" can represent the output number, or function value. x is then the independent variable, and y is the dependent variable.
Science and engineering use functions to describe and predict the behavior of the phenomena they deal with. Most naturally occurring phenomena are describable in terms of three general classes of functions: trigonometric, exponentials, and polynomials.
The natural trigonometric functions sine (sin) and cosine (cos) are used to construct functions that model and describe the behavior of oscillatory phenomena. In the following examples the independent variable is time, t.
1. Pendulum motion: .theta.=.theta..sub.m cos kt PA1 2. Simple business cycle model: i=i.sub.a +i.sub.v sin kt PA1 1. World population growth: P=P.sub.o e.sup.0.02(t-t.sbsp.o.sup.) (p=population at year t, p.sub.o =population at year t.sub.o.) PA1 2. Compound interest: ##EQU1## (A=accumulated value after t years, P=initial investment, r=annual rate, m=number of times compounded per year.) PA1 1. Falling object: y=y.sub.o -16t.sup.2 PA1 1. AM radio wave: V=A(1+m cos k.sub.1 t)(sin k.sub.2 t) PA1 2. FM radio wave: V=A sin (k.sub.1 t+m sin k.sub.2 t) PA1 3. Damped vibration: A=A.sub.o e.sup.-k.sbsp.1.sup.t cos k.sub.2 t
(.theta.=angle at time t, .theta..sub.m =maximum value of .theta., k=frequency coefficient.)
(i=interest rate at time t, i.sub.a =average interest rate, i.sub.v =maximum variation from the average, k=frequency coefficient.)
Many natural processes in which the rate of increase is itself increasing, or the rate of decrease is decreasing, follow exponential laws:
In each of the preceding examples the coefficients, i.e. the symbols which represent fixed values, have specific, meaningful "real world" interpretations.
A third class of functions is the polynomials. A polynomial is the sum of terms each consisting of a power of the independent variables multiplied by a constant coefficient, e.g. x.sup.2 +5x; 3x.sup.4 ; t.sup.3 -2t+1. Polynomials are more in the nature of "man-made" functions than the preceding two classes, but there are natural phenomena that fall into this class:
(y=height at time t, y.sub.o =initial height at t=0.)
The main significance of polynomials is that they are the easiest type of function to work with algebraically, and hence are normally the first type taught in high school and college mathematics courses, and are used to introduce the ideas of calculus.
Sometimes basic functions of one type, or of different types, are combined to form more complex functions:
In high school algebra courses students typically deal almost exclusively with polynomials, spending much time on the linear function, y=mx+b, and later the quadratic function y=ax.sup.2 +bx+c. Higher order polynomials are sometimes introduced after becoming familiar with the first and second order ones in detail. Exponentials and trigonometric functions are becoming more common in pre-calculus algebra courses, and are a significant part of the standard college calculus sequence.
In algebra and pre-calculus courses graphs are emphasized as the best way for students to get a "feel" for the basic functions. They are taught to correlate the various functions with their graphs: they first learn the general shape of the graph of each function type, then they learn how each coefficient specifically affects the shape. For example y=mx+b is always a straight line, m is the slope of the line, and b is the y-axis intercept. Similarly ax.sup.2 +bx+c is always a parabola, with the coefficients a, b and c each having specific influences on the appearance of the parabola.
In considering the prior art the most practical method of implementing the teaching system under consideration here is by means of an analog computer system together with an oscilloscope display. The use of an analog computer for solving differential equations or for simulation, with the solution displayed repetitively on an oscilloscope screen, is a commonly used technique. However standard analog computers, while well-suited to these two types of tasks, not suitable for the purpose of function generation in an educational setting. This is primarily a result of the requirement that the only programming necessary should be selecting the function type and setting coefficient values (as well as selecting between various optional features, if such choices exist).
To illustrate the problem involved, consider the analog computer implementation of the simple exponential function y=Ae.sup.kx. In an analog computer each integrator is given an initial value before computation. Normally the desired initial value is entered by setting a potentiometer (one for each integrator), and when the computer is placed in Reset mode, this value appears at the output of the integrator. In this example, as indicated in the diagrams, the integrator must be initialized to the initial function value Ae.sup.kx.sbsp.o, that is, to the value at x=x.sub.o. If x.sub.o is not zero, this number will be different for different functions (i.e. different values of A or k), and must be computed manually for each function before the initial condition can be set.
The same problem exists for other function types, and the fact that the computation must be done each time a coefficient is changed makes standard analog computers unusable for the educational purpose under consideration here.
A secondary problem is that standard analog computers are not constructed so that the potentiometer settings are easily visible from a distance.
The system could be built, in principle, with an "on-line" digital computer, but the cost and complexity of this approach would vastly exceed that of the analog computer approach.